Visual observation of photonic Floquet-Bloch oscillations

Bloch oscillations (BOs), a fundamental phenomenon of coherent quantum motion, describe the oscillatory movement of a quantum particle with a period *A*_BO in a crystal under a constant electric field. Initially observed in semiconductor superlattices and ultracold atoms, BOs have been extended to various wave systems. While extensively studied in static systems, recent interest has shifted to periodically driven (Floquet) systems due to their unique characteristics, leading to investigations of quasi-Bloch oscillations (QBOs) and super-Bloch oscillations (SBOs). QBOs occur when *A*_BO is an integer multiple of the modulation period *A*_FL, while SBOs arise when *A*_FL is approximately an integer multiple of *A*_BO. However, the connection between these remains unclear, and a general theory for BOs in Floquet systems is needed. Visual observation in Floquet systems is experimentally challenging due to the rapid temporal evolution of the wavefunction. This paper leverages the "photonic analogy", simulating temporal evolution through spatial light evolution in a waveguide array, to address this challenge. The authors present a general theory of BOs in photonic Floquet lattices and experimentally demonstrate FBOs, unifying QBOs and SBOs as special cases.

The paper reviews the extensive literature on Bloch oscillations, starting from their theoretical prediction by Bloch and Zener and their subsequent observation in various systems like semiconductor superlattices, ultracold atoms, acoustic cavities, and waveguide arrays. It highlights the recent focus on exploring Bloch oscillations in periodically driven (Floquet) systems, discussing the previously known types of Bloch-like oscillations – quasi-Bloch oscillations (QBOs) and super-Bloch oscillations (SBOs). The limitations of these previous studies, specifically the lack of a unifying theory and the experimental challenges in visually observing these oscillations in Floquet systems, are clearly stated. The paper then introduces the concept of photonic analogy as a powerful tool to overcome these experimental limitations, citing relevant works that employed this approach for simulating temporal evolution in quantum systems using spatial light evolution in waveguide arrays.

The authors utilize a femtosecond-laser-written waveguide array in a fused silica substrate as their experimental platform. A curved photonic lattice with a combined bending trajectory, consisting of a circular bending term (*x*_BO(z)) and a periodic bending term (*x*_FL(z)), is employed. The propagation of light in this lattice is described by a Schrödinger-type equation. By transforming to a reference frame where the waveguides are straight, an additional term F(z) appears in the equation, representing the effect of the combined trajectory. This term is crucial for simulating the electric field in the electronic analogy. The equation is then simplified using the nearest-neighbour tight-binding approximation, resulting in a set of coupled equations for the amplitudes of the guided modes in each waveguide. The generalized acceleration theory is applied to analyze the effect of the force term F(z) on the transverse Bloch momentum, resulting in a shift of the momentum and a modified propagation constant. The authors show that under specific conditions (when the ratio of A_FL/A_BO is a non-integer rational number), the period of the oscillations becomes the extended least common multiple (LCM) of A_FL and A_BO, thus defining the FBOs. To experimentally verify their findings, they fabricate samples with specific waveguide parameters and modulation periods and employ waveguide fluorescence microscopy to visualize the light evolution. The variance of excitation and the weighted average position of excitation are used to quantitatively analyze the breathing and oscillatory modes, respectively. Digital image processing involving coordinate transformation is used to enhance the visualization of the light evolution.

The paper presents the first visual observation of photonic Floquet-Bloch oscillations (FBOs) using waveguide fluorescence microscopy. The experiments directly visualize both the breathing and oscillatory motions of FBOs, providing detailed information on their evolution. The results demonstrate that the FBO period is determined by the extended least common multiple of the Floquet modulation period and the Bloch oscillation period. The authors show that for specific ratios of the modulation and Bloch oscillation periods, the FBOs degenerate into previously known QBOs and SBOs, thereby providing a unifying framework. Furthermore, two exotic properties of FBOs are experimentally verified: a fractal spectrum, where the FBO period is a Thomae's function of the ratio A_BO/A_FL, and fractional Floquet tunneling, where the FBO amplitude follows a linear combination of fractional-order Anger and Weber functions, deviating from the conventional integral-order Bessel function dependence. The experiments clearly show the sub-oscillations in the light evolution, a characteristic feature of FBOs not previously observed experimentally. The quantitative analysis of the variance of excitation and the weighted average position of excitation confirms the theoretical predictions for different ratios of A_BO/A_FL, showcasing the unique transport properties of FBOs, including their extended periods and dramatic broadening for non-integer ratios of A_BO/A_FL and ballistic spreading for integer ratios.

The findings of this paper significantly advance our understanding of Bloch oscillations in Floquet systems. By providing a general theory and experimental demonstration of FBOs, the authors unify previously disparate observations of QBOs and SBOs under a single framework. The experimental visualization of the FBOs, particularly the sub-oscillations, offers crucial insight into the underlying transport mechanism. The observed fractal spectrum and fractional Floquet tunneling further highlight the exotic properties of FBOs, expanding the scope of wave manipulation techniques. The photonic analogy approach offers a powerful experimental tool for studying Floquet systems, enabling detailed visualization of otherwise elusive dynamics. The results have broad implications for various fields, such as photonics, where the precise control of light propagation is crucial, and condensed matter physics and quantum physics, where the understanding of quantum transport phenomena is essential.

This research successfully demonstrates the first visual observation of photonic Floquet-Bloch oscillations (FBOs), unifying previous observations of QBOs and SBOs. The experimental findings confirm the theoretical predictions, revealing unique properties such as a fractal spectrum and fractional Floquet tunneling. The photonic analogy approach employed here provides a valuable platform for studying Floquet systems. Future research could explore the application of FBOs in developing novel photonic devices and further investigating the complex dynamics of quantum transport in periodically driven systems.

While the study provides a comprehensive analysis of FBOs, potential limitations include the use of the nearest-neighbour tight-binding approximation, which simplifies the system's dynamics. The experimental setup, while innovative, might be limited by the fabrication precision of the waveguide array and the accuracy of the fluorescence microscopy measurements. Further investigations could explore the influence of higher-order interactions and imperfections in the waveguide structures on the observed FBOs.